Numerical treatment of non-integer order partial differential equations by omitting discretization of data

Abstract

In this manuscript, we derived the shifted Jacobi operational matrices of fractional derivatives and integration which are applied for numerical solution of general linear multi-term fractional partial differential equations (FPDEs). A new approach implementing shifted Jacobi operational matrix without using the shifted Jacobi collocation technique is introduced for the numerical solution of multi-term FPDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. Further the proposed method needs no discretization of data. The proposed method is applied for solving linear multi-term FPDEs subject to initial conditions and the approximate solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the considered method with those found by other known methods like homotopy analysis (HAM) method. For computation purposes, we use Matlab 2016.

Appendices

Appendix A

In this appendix we recall some well-known definitions and notions.

Definition 4.1

Hilfer (2000) The Riemann–Liouville integral of arbitrary order $p> 0$ of a function f(t) is defined by

$$\begin{aligned} {\mathcal {I}}^{p}f(t)=\frac{1}{\Gamma {(p)}}\int _{0}^{t}(t-\tau )^{p-1}f(\tau )d\tau , \end{aligned}$$

such that the integral on right-hand side converges pointwise on $(0, \infty )$. Also the aforesaid integral satisfies the following relations:

• (1) ${\mathcal {I}}^0 f(t)= f(t);$

• (2) ${\mathcal {I}}^p {\mathcal {I}}^q f(t)={\mathcal {I}}^{p+q} f(t);$

• (3) ${\mathcal {I}}^p t^\gamma =\frac{\Gamma (\gamma +1)}{\Gamma (p+\gamma +1)}t^{p+\gamma }.$

Definition 4.2

Hilfer (2000) For a given function f(t), the Caputo fractional order derivative of order p is defined as

$$\begin{aligned} {\mathcal {D}}^p f(t)=\frac{1}{\Gamma (k-p)}\int _0^{t}\frac{f^{(k)}(s)}{(t-\tau )^{p+1-k}}d\tau , \ k-1< p \le k \ ,\, k=[\alpha ]+1, \end{aligned}$$

such that the right side is pointwise defined on $(0,\infty )$. Further, the operator ${\mathcal {D}}$ satisfies the following properties in particular for any constant C, ${\mathcal {D}}^{p}C=0$ and

$$\begin{aligned} {\mathcal {D}}^{p}t^i=\left\{ \begin{aligned}&0,\ i\in N,\ i< p,\\&\frac{\Gamma (1+i)}{\Gamma (1+i-p)}t^{i-p}, i\ge p.\end{aligned}\right. \end{aligned}$$

The following relations are necessary throughout this paper:

Theorem 4.3

Hilfer (2000) For any function V(t), the following results hold:

• (a) ${\mathcal {I}}^{\alpha }[{\mathcal {D}}^{p}V(t)]=0$ implies that $V(t)= \sum _{i=0}^{k-1}V^{(i)}\frac{t^i}{\Gamma (i+1)};$

• (b) ${\mathcal {I}}^{p}{\mathcal {D}}^{\alpha }V(t)=V(t)-\sum _{i=0}^{k-1}V^{(i)}\frac{t^i}{\Gamma (i+1)};$

• (c) ${\mathcal {D}}^{p}(\lambda U(t)+\mu V(t))=\lambda {\mathcal {D}}^{\alpha }U(t)+\mu {\mathcal {D}}^{p}V(t).$

Appendix B

Next the proof of Theorem 2.3 is provided below.

Proof

In order to prove the result, take $\digamma _{L,n}^{(\eta ,\xi )}(t,x)$ as defined by (8), then the fractional integral of order p of $\digamma _{L,n}^{(\eta ,\xi )}(t,x)$ with respect to t is given by the relation

$$\begin{aligned} \begin{aligned}&{\mathcal {I}}_t^p \digamma _{L,n}^{(\eta ,\xi )}(t,x)={\mathcal {I}}_t^p \digamma _{L,a}^{(\eta ,\xi )}(t)\digamma _{L,b}^{(\eta ,\xi )}(x)\\&\quad =\sum _{n=0}^a \frac{(-1)^{a-n}\Gamma (a+\xi +1)\Gamma (a+n+\eta +\beta +1)}{\Gamma (n+\xi +1)\Gamma (a+\eta +\xi +1)\Gamma (a-n+1)\Gamma (n+1)L^n}{\mathcal {I}}_t^p t^{n}\digamma _{L,a}^{(\eta ,\xi )}(x), \end{aligned} \end{aligned}$$

which in view of the definition of fractional integrals takes the following form:

$$\begin{aligned} {\mathcal {I}}_t^p&\digamma _{L,a}^{(\eta ,\xi )}(t)\digamma _{L,b}^{(\eta ,\xi )}(x)\nonumber \\&\quad =\sum _{n=0}^a \frac{(-1)^{a-n}\Gamma (a+\xi +1)\Gamma (a+n+\eta +\xi +1)\Gamma (1+n)}{\Gamma (n+\eta +1)\Gamma (a+\xi +\eta +1)\Gamma (a-n+1)\Gamma (n+1)\Gamma (1+n+p)L^n}\nonumber \\&\qquad \times t^{n+p}\digamma _{L,b}^{(\eta ,\xi )}(x). \end{aligned}$$

(41)

Approximating $t^{n+p}\digamma _{L,b}^{(\eta ,\xi )}(x)$ interm of shifted Jacobi polynomials of two variables, we obtain

$$\begin{aligned} t^{n+p}\digamma _{L,b}^{(\eta ,\xi )}(x)\approx \sum _{i=0}^m\sum _{j=0}^mS_{i,j,b}\digamma _{L,i}^{(\eta ,\xi )}(t)\digamma _{L,j}^{(\eta ,\xi )}(x), \end{aligned}$$

(42)

where $S_{i,j,b}=\frac{\delta _{j,b}}{{\mathfrak {R}}_{L,i}^{(\eta ,\xi )},{\mathfrak {R}}_{L,j}^{(\eta ,\xi )}}\int _0^{L}\int _0^{L}t^{n+p}\digamma _{L,b}^{(\eta ,\xi )}(x)\digamma _{L,i}^{(\eta ,\xi )}(t)\digamma _{L,j}^{(\eta ,\xi )}(x){\mathcal {W}}_{L}^{(\eta ,\xi )}(t,x)\mathrm{d}t\mathrm{d}x.$ Which further in view of orthogonality relation as in Doha et al. (2012) implies that

$$\begin{aligned} S_{i,j,b}=\frac{\delta _{j,b}}{{\mathfrak {R}}_{L,i}^{(\eta ,\xi )}}\int _0^{L}t^{n+p}\digamma _{L,i}^{(\eta ,\xi )}(x){\mathcal {W}}_{L}^{(\eta ,\xi )}(t)dt. \end{aligned}$$

(43)

Which on using (4),(2) yields

$$\begin{aligned} S_{i,j,b}= & {} \frac{\delta _{j,b}}{{\mathfrak {R}}_{L,i}^{(\eta ,\xi )}}\sum _{l=0}^i \frac{(-1)^{i-l}\Gamma (i+\xi +1)\Gamma (i+l+\eta +\xi +1)}{\Gamma (l+\xi +1)\Gamma (i+\eta +\xi +1)\Gamma (i-l+1)\Gamma (l+1)L^l}\nonumber \\&\int _0^{L}t^{n+p+l+\xi } (L-t)^{p}dt. \end{aligned}$$

(44)

Using the Convolution theorem of Laplace transformation in (44), we have

$$\begin{aligned} \pounds \{\int _0^{L}t^{n+p+l+\xi } (L-t)^{\eta }dt\}=\frac{\Gamma (n+p+l+\xi +1)\Gamma (\eta +1)}{s^{(n+p+l+\eta +\xi +2)}}. \end{aligned}$$

(45)

Taking inverse laplace of (45), we get

$$\begin{aligned} \int _0^{L}t^{n+p+l+\xi } (L-t)^{\eta }dt=\frac{\Gamma (n+p+l+\xi +1)\Gamma (\eta +1)L^{(n+p+l+\xi +\eta +1)}}{\Gamma (n+p+l+\xi +\eta +1)}. \end{aligned}$$

(46)

Therefore, one can get the coefficient of (42) as

$$\begin{aligned}&S_{ijb}\nonumber \\&\quad =\frac{\delta _{j,b}}{{\mathfrak {R}}_{L,i}^{(\eta ,\xi )}}\sum _{l=0}^i \frac{(-1)^{i-l}\Gamma (i+\xi +1)\Gamma (i+l+\eta +\xi +1)\Gamma (n+p+l+\xi +1)\Gamma (\eta +1)L^{(n+p+l+\xi +\eta +1)}}{\Gamma (l+\xi +1)\Gamma (i+\eta +\xi +1)\Gamma (i-l+1)\Gamma (l+1)L^l\Gamma (n+p+l+\xi +\eta +2)}.\nonumber \\ \end{aligned}$$

(47)

Using(5) and simplifying we get the generalized value as

$$\begin{aligned}&S_{ijb}\nonumber \\&\quad =\delta _{j,b}\sum _{l=0}^i \frac{(-1)^{i-l}(2i+\eta +\xi +1)\Gamma (i+1)\Gamma (i+l+\eta +\xi +1)\Gamma (n+p+l+\xi +1)\Gamma (\eta +1)L^{p}}{\Gamma (i+\eta +1)\Gamma (l+\xi +1)\Gamma (i-l+1)\Gamma (l+1)\Gamma (n+p+l+\xi +\eta +2)}\nonumber \\ \end{aligned}$$

(48)

For simplicity of notation let

$$\begin{aligned} \Delta _{a,n,p}=\frac{(-1)^{a-n}\Gamma (a+\xi +1)\Gamma (a+n+\eta +\xi +1)\Gamma (1+n)}{\Gamma (n+\xi +1)\Gamma (a+\eta +\xi +1)\Gamma (a-n+1)\Gamma (n+1)\Gamma (1+n+p)L^n}.\nonumber \\ \end{aligned}$$

(49)

Plugging (48),(49)(42) in (41), we get

$$\begin{aligned} {\mathcal {I}}_t^p \digamma _{L,a}^{(\eta ,\xi )}(t)\digamma _{L,b}^{(\eta ,\xi )}(x) =\sum _{n=0}^a\Lambda _{a,k,p}\sum _{i=0}^m\sum _{j=0}^mS_{i,j,b}\digamma _{L,i}^{(\eta ,\xi )}(t)\digamma _{L,j}^{(\eta ,\xi )}(x). \end{aligned}$$

(50)

The above relation (50) can also be written as

$$\begin{aligned} {\mathcal {I}}_t^p \digamma _{L,a}^{(\eta ,\xi )}(t)\digamma _{L,b}^{(\eta ,\xi )}(x) =\sum _{i=0}^m\sum _{j=0}^m\sum _{k=0}^a\Delta _{a,k,p}G_{i,j,b}\digamma _{L,i}^{(\eta ,\xi )}(t)\digamma _{L,j}^{(\eta ,\xi )}(x). \end{aligned}$$

(51)

Using

$$\begin{aligned} \Omega _{i,j,a,b,n}=\sum _{n=0}^a\Delta _{a,n,p}G_{i,j,b}, \end{aligned}$$

(52)

one has

$$\begin{aligned} {\mathcal {I}}^p \digamma _{L,a}^{(\eta ,\xi )}(t)\digamma _{L,b}^{(\eta ,\xi )}(x) =\sum _{i=0}^m\sum _{j=0}^m\Omega _{i,j,a,b,n}\digamma _{L,i}^{(\eta ,\xi )}(t)\digamma _{L,j}^{(\eta ,\xi )}(x). \end{aligned}$$

(53)

Also using $r=Ki+j+1,\, v=Ka+b+1,\,\prod _{v,r,n}=\Omega _{i,j,b,a,n} \text { for } i,\,j,\,a,\,b=0,1,2,3,...,m$, we get the required result. $\square$